Properties

Label 38.8.a.d.1.1
Level $38$
Weight $8$
Character 38.1
Self dual yes
Analytic conductor $11.871$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.8706309684\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{633}) \)
Defining polynomial: \(x^{2} - x - 158\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(13.0797\) of defining polynomial
Character \(\chi\) \(=\) 38.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} -72.2392 q^{3} +64.0000 q^{4} +467.472 q^{5} -577.914 q^{6} -1471.23 q^{7} +512.000 q^{8} +3031.51 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -72.2392 q^{3} +64.0000 q^{4} +467.472 q^{5} -577.914 q^{6} -1471.23 q^{7} +512.000 q^{8} +3031.51 q^{9} +3739.78 q^{10} -3547.04 q^{11} -4623.31 q^{12} -12361.8 q^{13} -11769.9 q^{14} -33769.8 q^{15} +4096.00 q^{16} -13486.3 q^{17} +24252.1 q^{18} -6859.00 q^{19} +29918.2 q^{20} +106281. q^{21} -28376.3 q^{22} -94084.9 q^{23} -36986.5 q^{24} +140405. q^{25} -98894.4 q^{26} -61006.6 q^{27} -94158.9 q^{28} +33541.0 q^{29} -270159. q^{30} +212096. q^{31} +32768.0 q^{32} +256236. q^{33} -107890. q^{34} -687760. q^{35} +194016. q^{36} +33656.5 q^{37} -54872.0 q^{38} +893007. q^{39} +239346. q^{40} -480531. q^{41} +850246. q^{42} -561340. q^{43} -227011. q^{44} +1.41715e6 q^{45} -752679. q^{46} +478445. q^{47} -295892. q^{48} +1.34098e6 q^{49} +1.12324e6 q^{50} +974237. q^{51} -791156. q^{52} -93127.0 q^{53} -488053. q^{54} -1.65814e6 q^{55} -753271. q^{56} +495489. q^{57} +268328. q^{58} +955034. q^{59} -2.16127e6 q^{60} -1.71632e6 q^{61} +1.69677e6 q^{62} -4.46005e6 q^{63} +262144. q^{64} -5.77880e6 q^{65} +2.04988e6 q^{66} +476255. q^{67} -863120. q^{68} +6.79662e6 q^{69} -5.50208e6 q^{70} +1.95518e6 q^{71} +1.55213e6 q^{72} +1.21632e6 q^{73} +269252. q^{74} -1.01428e7 q^{75} -438976. q^{76} +5.21852e6 q^{77} +7.14406e6 q^{78} +3.94544e6 q^{79} +1.91477e6 q^{80} -2.22284e6 q^{81} -3.84425e6 q^{82} -3.54252e6 q^{83} +6.80197e6 q^{84} -6.30445e6 q^{85} -4.49072e6 q^{86} -2.42298e6 q^{87} -1.81609e6 q^{88} -7.50202e6 q^{89} +1.13372e7 q^{90} +1.81871e7 q^{91} -6.02143e6 q^{92} -1.53216e7 q^{93} +3.82756e6 q^{94} -3.20639e6 q^{95} -2.36714e6 q^{96} +1.27974e7 q^{97} +1.07279e7 q^{98} -1.07529e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 16q^{2} - 69q^{3} + 128q^{4} + 155q^{5} - 552q^{6} - 2238q^{7} + 1024q^{8} + 855q^{9} + O(q^{10}) \) \( 2q + 16q^{2} - 69q^{3} + 128q^{4} + 155q^{5} - 552q^{6} - 2238q^{7} + 1024q^{8} + 855q^{9} + 1240q^{10} - 3295q^{11} - 4416q^{12} - 13427q^{13} - 17904q^{14} - 34782q^{15} + 8192q^{16} - 32256q^{17} + 6840q^{18} - 13718q^{19} + 9920q^{20} + 103797q^{21} - 26360q^{22} - 82525q^{23} - 35328q^{24} + 159919q^{25} - 107416q^{26} - 75141q^{27} - 143232q^{28} - 12749q^{29} - 278256q^{30} + 258944q^{31} + 65536q^{32} + 257052q^{33} - 258048q^{34} - 448167q^{35} + 54720q^{36} - 149260q^{37} - 109744q^{38} + 889557q^{39} + 79360q^{40} + 339130q^{41} + 830376q^{42} - 83869q^{43} - 210880q^{44} + 2097243q^{45} - 660200q^{46} + 1471025q^{47} - 282624q^{48} + 1105372q^{49} + 1279352q^{50} + 913437q^{51} - 859328q^{52} - 945643q^{53} - 601128q^{54} - 1736899q^{55} - 1145856q^{56} + 473271q^{57} - 101992q^{58} - 969009q^{59} - 2226048q^{60} - 1506755q^{61} + 2071552q^{62} - 2791179q^{63} + 524288q^{64} - 5445956q^{65} + 2056416q^{66} - 1848219q^{67} - 2064384q^{68} + 6834063q^{69} - 3585336q^{70} - 3417184q^{71} + 437760q^{72} - 2499822q^{73} - 1194080q^{74} - 10079553q^{75} - 877952q^{76} + 5025267q^{77} + 7116456q^{78} + 2636926q^{79} + 634880q^{80} + 2491398q^{81} + 2713040q^{82} - 10059354q^{83} + 6643008q^{84} - 439425q^{85} - 670952q^{86} - 2572923q^{87} - 1687040q^{88} - 3506160q^{89} + 16777944q^{90} + 19003851q^{91} - 5281600q^{92} - 15169884q^{93} + 11768200q^{94} - 1063145q^{95} - 2260992q^{96} + 5893526q^{97} + 8842976q^{98} - 11301453q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) −72.2392 −1.54472 −0.772358 0.635187i \(-0.780923\pi\)
−0.772358 + 0.635187i \(0.780923\pi\)
\(4\) 64.0000 0.500000
\(5\) 467.472 1.67248 0.836240 0.548364i \(-0.184749\pi\)
0.836240 + 0.548364i \(0.184749\pi\)
\(6\) −577.914 −1.09228
\(7\) −1471.23 −1.62121 −0.810603 0.585596i \(-0.800861\pi\)
−0.810603 + 0.585596i \(0.800861\pi\)
\(8\) 512.000 0.353553
\(9\) 3031.51 1.38615
\(10\) 3739.78 1.18262
\(11\) −3547.04 −0.803511 −0.401756 0.915747i \(-0.631600\pi\)
−0.401756 + 0.915747i \(0.631600\pi\)
\(12\) −4623.31 −0.772358
\(13\) −12361.8 −1.56056 −0.780279 0.625431i \(-0.784923\pi\)
−0.780279 + 0.625431i \(0.784923\pi\)
\(14\) −11769.9 −1.14637
\(15\) −33769.8 −2.58351
\(16\) 4096.00 0.250000
\(17\) −13486.3 −0.665764 −0.332882 0.942969i \(-0.608021\pi\)
−0.332882 + 0.942969i \(0.608021\pi\)
\(18\) 24252.1 0.980155
\(19\) −6859.00 −0.229416
\(20\) 29918.2 0.836240
\(21\) 106281. 2.50430
\(22\) −28376.3 −0.568168
\(23\) −94084.9 −1.61240 −0.806199 0.591644i \(-0.798479\pi\)
−0.806199 + 0.591644i \(0.798479\pi\)
\(24\) −36986.5 −0.546140
\(25\) 140405. 1.79719
\(26\) −98894.4 −1.10348
\(27\) −61006.6 −0.596490
\(28\) −94158.9 −0.810603
\(29\) 33541.0 0.255378 0.127689 0.991814i \(-0.459244\pi\)
0.127689 + 0.991814i \(0.459244\pi\)
\(30\) −270159. −1.82681
\(31\) 212096. 1.27869 0.639346 0.768919i \(-0.279205\pi\)
0.639346 + 0.768919i \(0.279205\pi\)
\(32\) 32768.0 0.176777
\(33\) 256236. 1.24120
\(34\) −107890. −0.470766
\(35\) −687760. −2.71143
\(36\) 194016. 0.693074
\(37\) 33656.5 0.109235 0.0546176 0.998507i \(-0.482606\pi\)
0.0546176 + 0.998507i \(0.482606\pi\)
\(38\) −54872.0 −0.162221
\(39\) 893007. 2.41062
\(40\) 239346. 0.591311
\(41\) −480531. −1.08887 −0.544437 0.838801i \(-0.683257\pi\)
−0.544437 + 0.838801i \(0.683257\pi\)
\(42\) 850246. 1.77081
\(43\) −561340. −1.07668 −0.538339 0.842728i \(-0.680948\pi\)
−0.538339 + 0.842728i \(0.680948\pi\)
\(44\) −227011. −0.401756
\(45\) 1.41715e6 2.31830
\(46\) −752679. −1.14014
\(47\) 478445. 0.672187 0.336093 0.941829i \(-0.390894\pi\)
0.336093 + 0.941829i \(0.390894\pi\)
\(48\) −295892. −0.386179
\(49\) 1.34098e6 1.62831
\(50\) 1.12324e6 1.27080
\(51\) 974237. 1.02842
\(52\) −791156. −0.780279
\(53\) −93127.0 −0.0859232 −0.0429616 0.999077i \(-0.513679\pi\)
−0.0429616 + 0.999077i \(0.513679\pi\)
\(54\) −488053. −0.421782
\(55\) −1.65814e6 −1.34386
\(56\) −753271. −0.573183
\(57\) 495489. 0.354382
\(58\) 268328. 0.180579
\(59\) 955034. 0.605392 0.302696 0.953087i \(-0.402113\pi\)
0.302696 + 0.953087i \(0.402113\pi\)
\(60\) −2.16127e6 −1.29175
\(61\) −1.71632e6 −0.968152 −0.484076 0.875026i \(-0.660844\pi\)
−0.484076 + 0.875026i \(0.660844\pi\)
\(62\) 1.69677e6 0.904172
\(63\) −4.46005e6 −2.24723
\(64\) 262144. 0.125000
\(65\) −5.77880e6 −2.61000
\(66\) 2.04988e6 0.877659
\(67\) 476255. 0.193454 0.0967270 0.995311i \(-0.469163\pi\)
0.0967270 + 0.995311i \(0.469163\pi\)
\(68\) −863120. −0.332882
\(69\) 6.79662e6 2.49070
\(70\) −5.50208e6 −1.91727
\(71\) 1.95518e6 0.648311 0.324155 0.946004i \(-0.394920\pi\)
0.324155 + 0.946004i \(0.394920\pi\)
\(72\) 1.55213e6 0.490078
\(73\) 1.21632e6 0.365948 0.182974 0.983118i \(-0.441428\pi\)
0.182974 + 0.983118i \(0.441428\pi\)
\(74\) 269252. 0.0772409
\(75\) −1.01428e7 −2.77614
\(76\) −438976. −0.114708
\(77\) 5.21852e6 1.30266
\(78\) 7.14406e6 1.70457
\(79\) 3.94544e6 0.900328 0.450164 0.892946i \(-0.351366\pi\)
0.450164 + 0.892946i \(0.351366\pi\)
\(80\) 1.91477e6 0.418120
\(81\) −2.22284e6 −0.464740
\(82\) −3.84425e6 −0.769951
\(83\) −3.54252e6 −0.680048 −0.340024 0.940417i \(-0.610435\pi\)
−0.340024 + 0.940417i \(0.610435\pi\)
\(84\) 6.80197e6 1.25215
\(85\) −6.30445e6 −1.11348
\(86\) −4.49072e6 −0.761327
\(87\) −2.42298e6 −0.394487
\(88\) −1.81609e6 −0.284084
\(89\) −7.50202e6 −1.12801 −0.564006 0.825771i \(-0.690740\pi\)
−0.564006 + 0.825771i \(0.690740\pi\)
\(90\) 1.13372e7 1.63929
\(91\) 1.81871e7 2.52999
\(92\) −6.02143e6 −0.806199
\(93\) −1.53216e7 −1.97522
\(94\) 3.82756e6 0.475308
\(95\) −3.20639e6 −0.383693
\(96\) −2.36714e6 −0.273070
\(97\) 1.27974e7 1.42371 0.711855 0.702326i \(-0.247855\pi\)
0.711855 + 0.702326i \(0.247855\pi\)
\(98\) 1.07279e7 1.15139
\(99\) −1.07529e7 −1.11379
\(100\) 8.98593e6 0.898593
\(101\) 332855. 0.0321463 0.0160731 0.999871i \(-0.494884\pi\)
0.0160731 + 0.999871i \(0.494884\pi\)
\(102\) 7.79389e6 0.727200
\(103\) 4.52517e6 0.408042 0.204021 0.978967i \(-0.434599\pi\)
0.204021 + 0.978967i \(0.434599\pi\)
\(104\) −6.32924e6 −0.551741
\(105\) 4.96833e7 4.18840
\(106\) −745016. −0.0607568
\(107\) 7.33885e6 0.579141 0.289571 0.957157i \(-0.406487\pi\)
0.289571 + 0.957157i \(0.406487\pi\)
\(108\) −3.90442e6 −0.298245
\(109\) −1.03920e7 −0.768612 −0.384306 0.923206i \(-0.625559\pi\)
−0.384306 + 0.923206i \(0.625559\pi\)
\(110\) −1.32651e7 −0.950249
\(111\) −2.43132e6 −0.168737
\(112\) −6.02617e6 −0.405302
\(113\) 5.33971e6 0.348131 0.174066 0.984734i \(-0.444309\pi\)
0.174066 + 0.984734i \(0.444309\pi\)
\(114\) 3.96391e6 0.250586
\(115\) −4.39820e7 −2.69670
\(116\) 2.14663e6 0.127689
\(117\) −3.74749e7 −2.16317
\(118\) 7.64027e6 0.428077
\(119\) 1.98414e7 1.07934
\(120\) −1.72902e7 −0.913407
\(121\) −6.90567e6 −0.354370
\(122\) −1.37306e7 −0.684587
\(123\) 3.47132e7 1.68200
\(124\) 1.35741e7 0.639346
\(125\) 2.91142e7 1.33328
\(126\) −3.56804e7 −1.58903
\(127\) −3.43331e7 −1.48730 −0.743652 0.668566i \(-0.766908\pi\)
−0.743652 + 0.668566i \(0.766908\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 4.05507e7 1.66316
\(130\) −4.62304e7 −1.84555
\(131\) 250925. 0.00975202 0.00487601 0.999988i \(-0.498448\pi\)
0.00487601 + 0.999988i \(0.498448\pi\)
\(132\) 1.63991e7 0.620598
\(133\) 1.00912e7 0.371930
\(134\) 3.81004e6 0.136793
\(135\) −2.85189e7 −0.997618
\(136\) −6.90496e6 −0.235383
\(137\) −2.54613e7 −0.845978 −0.422989 0.906135i \(-0.639019\pi\)
−0.422989 + 0.906135i \(0.639019\pi\)
\(138\) 5.43729e7 1.76119
\(139\) 4.33237e7 1.36828 0.684138 0.729352i \(-0.260178\pi\)
0.684138 + 0.729352i \(0.260178\pi\)
\(140\) −4.40167e7 −1.35572
\(141\) −3.45625e7 −1.03834
\(142\) 1.56415e7 0.458425
\(143\) 4.38478e7 1.25393
\(144\) 1.24171e7 0.346537
\(145\) 1.56795e7 0.427114
\(146\) 9.73058e6 0.258764
\(147\) −9.68716e7 −2.51528
\(148\) 2.15401e6 0.0546176
\(149\) −3.98620e7 −0.987204 −0.493602 0.869688i \(-0.664320\pi\)
−0.493602 + 0.869688i \(0.664320\pi\)
\(150\) −8.11421e7 −1.96303
\(151\) −2.13810e6 −0.0505369 −0.0252684 0.999681i \(-0.508044\pi\)
−0.0252684 + 0.999681i \(0.508044\pi\)
\(152\) −3.51181e6 −0.0811107
\(153\) −4.08837e7 −0.922847
\(154\) 4.17482e7 0.921118
\(155\) 9.91489e7 2.13859
\(156\) 5.71525e7 1.20531
\(157\) −3.04702e7 −0.628387 −0.314193 0.949359i \(-0.601734\pi\)
−0.314193 + 0.949359i \(0.601734\pi\)
\(158\) 3.15635e7 0.636628
\(159\) 6.72743e6 0.132727
\(160\) 1.53181e7 0.295655
\(161\) 1.38421e8 2.61403
\(162\) −1.77827e7 −0.328621
\(163\) −9.48952e7 −1.71628 −0.858139 0.513417i \(-0.828379\pi\)
−0.858139 + 0.513417i \(0.828379\pi\)
\(164\) −3.07540e7 −0.544437
\(165\) 1.19783e8 2.07588
\(166\) −2.83402e7 −0.480867
\(167\) −1.10332e8 −1.83313 −0.916563 0.399890i \(-0.869048\pi\)
−0.916563 + 0.399890i \(0.869048\pi\)
\(168\) 5.44157e7 0.885405
\(169\) 9.00657e7 1.43534
\(170\) −5.04356e7 −0.787346
\(171\) −2.07931e7 −0.318004
\(172\) −3.59257e7 −0.538339
\(173\) 1.06112e8 1.55813 0.779066 0.626942i \(-0.215694\pi\)
0.779066 + 0.626942i \(0.215694\pi\)
\(174\) −1.93838e7 −0.278944
\(175\) −2.06569e8 −2.91361
\(176\) −1.45287e7 −0.200878
\(177\) −6.89909e7 −0.935159
\(178\) −6.00162e7 −0.797624
\(179\) 4.88351e7 0.636424 0.318212 0.948020i \(-0.396918\pi\)
0.318212 + 0.948020i \(0.396918\pi\)
\(180\) 9.06973e7 1.15915
\(181\) 5.52074e6 0.0692026 0.0346013 0.999401i \(-0.488984\pi\)
0.0346013 + 0.999401i \(0.488984\pi\)
\(182\) 1.45497e8 1.78897
\(183\) 1.23986e8 1.49552
\(184\) −4.81714e7 −0.570069
\(185\) 1.57335e7 0.182693
\(186\) −1.22573e8 −1.39669
\(187\) 4.78363e7 0.534948
\(188\) 3.06205e7 0.336093
\(189\) 8.97549e7 0.967034
\(190\) −2.56511e7 −0.271312
\(191\) −1.15562e8 −1.20005 −0.600023 0.799983i \(-0.704842\pi\)
−0.600023 + 0.799983i \(0.704842\pi\)
\(192\) −1.89371e7 −0.193090
\(193\) 4.01954e6 0.0402463 0.0201232 0.999798i \(-0.493594\pi\)
0.0201232 + 0.999798i \(0.493594\pi\)
\(194\) 1.02379e8 1.00672
\(195\) 4.17456e8 4.03171
\(196\) 8.58229e7 0.814155
\(197\) 8.92289e7 0.831522 0.415761 0.909474i \(-0.363515\pi\)
0.415761 + 0.909474i \(0.363515\pi\)
\(198\) −8.60231e7 −0.787566
\(199\) −1.73811e8 −1.56348 −0.781740 0.623604i \(-0.785668\pi\)
−0.781740 + 0.623604i \(0.785668\pi\)
\(200\) 7.18875e7 0.635401
\(201\) −3.44043e7 −0.298832
\(202\) 2.66284e6 0.0227308
\(203\) −4.93467e7 −0.414020
\(204\) 6.23511e7 0.514208
\(205\) −2.24635e8 −1.82112
\(206\) 3.62014e7 0.288529
\(207\) −2.85219e8 −2.23502
\(208\) −5.06340e7 −0.390140
\(209\) 2.43292e7 0.184338
\(210\) 3.97466e8 2.96164
\(211\) −1.48188e8 −1.08598 −0.542992 0.839738i \(-0.682709\pi\)
−0.542992 + 0.839738i \(0.682709\pi\)
\(212\) −5.96013e6 −0.0429616
\(213\) −1.41241e8 −1.00146
\(214\) 5.87108e7 0.409515
\(215\) −2.62411e8 −1.80072
\(216\) −3.12354e7 −0.210891
\(217\) −3.12042e8 −2.07302
\(218\) −8.31361e7 −0.543491
\(219\) −8.78662e7 −0.565285
\(220\) −1.06121e8 −0.671928
\(221\) 1.66714e8 1.03896
\(222\) −1.94505e7 −0.119315
\(223\) 1.38594e6 0.00836907 0.00418454 0.999991i \(-0.498668\pi\)
0.00418454 + 0.999991i \(0.498668\pi\)
\(224\) −4.82094e7 −0.286591
\(225\) 4.25639e8 2.49117
\(226\) 4.27177e7 0.246166
\(227\) 1.35147e7 0.0766858 0.0383429 0.999265i \(-0.487792\pi\)
0.0383429 + 0.999265i \(0.487792\pi\)
\(228\) 3.17113e7 0.177191
\(229\) 3.09882e7 0.170519 0.0852594 0.996359i \(-0.472828\pi\)
0.0852594 + 0.996359i \(0.472828\pi\)
\(230\) −3.51856e8 −1.90686
\(231\) −3.76982e8 −2.01224
\(232\) 1.71730e7 0.0902897
\(233\) −2.04142e8 −1.05727 −0.528635 0.848849i \(-0.677296\pi\)
−0.528635 + 0.848849i \(0.677296\pi\)
\(234\) −2.99799e8 −1.52959
\(235\) 2.23660e8 1.12422
\(236\) 6.11221e7 0.302696
\(237\) −2.85016e8 −1.39075
\(238\) 1.58731e8 0.763209
\(239\) −2.25595e8 −1.06890 −0.534449 0.845201i \(-0.679481\pi\)
−0.534449 + 0.845201i \(0.679481\pi\)
\(240\) −1.38321e8 −0.645876
\(241\) 4.16156e7 0.191512 0.0957562 0.995405i \(-0.469473\pi\)
0.0957562 + 0.995405i \(0.469473\pi\)
\(242\) −5.52453e7 −0.250577
\(243\) 2.93998e8 1.31438
\(244\) −1.09844e8 −0.484076
\(245\) 6.26872e8 2.72331
\(246\) 2.77706e8 1.18936
\(247\) 8.47896e7 0.358017
\(248\) 1.08593e8 0.452086
\(249\) 2.55909e8 1.05048
\(250\) 2.32914e8 0.942769
\(251\) 4.55579e8 1.81847 0.909233 0.416287i \(-0.136669\pi\)
0.909233 + 0.416287i \(0.136669\pi\)
\(252\) −2.85443e8 −1.12362
\(253\) 3.33723e8 1.29558
\(254\) −2.74665e8 −1.05168
\(255\) 4.55428e8 1.72000
\(256\) 1.67772e7 0.0625000
\(257\) −2.46264e8 −0.904972 −0.452486 0.891772i \(-0.649463\pi\)
−0.452486 + 0.891772i \(0.649463\pi\)
\(258\) 3.24406e8 1.17603
\(259\) −4.95165e7 −0.177093
\(260\) −3.69843e8 −1.30500
\(261\) 1.01680e8 0.353992
\(262\) 2.00740e6 0.00689572
\(263\) 4.04810e8 1.37217 0.686083 0.727523i \(-0.259329\pi\)
0.686083 + 0.727523i \(0.259329\pi\)
\(264\) 1.31193e8 0.438829
\(265\) −4.35343e7 −0.143705
\(266\) 8.07295e7 0.262994
\(267\) 5.41940e8 1.74246
\(268\) 3.04803e7 0.0967270
\(269\) −2.44584e8 −0.766118 −0.383059 0.923724i \(-0.625129\pi\)
−0.383059 + 0.923724i \(0.625129\pi\)
\(270\) −2.28151e8 −0.705422
\(271\) 7.88689e7 0.240721 0.120360 0.992730i \(-0.461595\pi\)
0.120360 + 0.992730i \(0.461595\pi\)
\(272\) −5.52397e7 −0.166441
\(273\) −1.31382e9 −3.90811
\(274\) −2.03691e8 −0.598196
\(275\) −4.98023e8 −1.44406
\(276\) 4.34984e8 1.24535
\(277\) −2.30329e8 −0.651133 −0.325567 0.945519i \(-0.605555\pi\)
−0.325567 + 0.945519i \(0.605555\pi\)
\(278\) 3.46590e8 0.967518
\(279\) 6.42970e8 1.77246
\(280\) −3.52133e8 −0.958637
\(281\) −5.44589e8 −1.46419 −0.732094 0.681203i \(-0.761457\pi\)
−0.732094 + 0.681203i \(0.761457\pi\)
\(282\) −2.76500e8 −0.734216
\(283\) −3.69887e7 −0.0970101 −0.0485051 0.998823i \(-0.515446\pi\)
−0.0485051 + 0.998823i \(0.515446\pi\)
\(284\) 1.25132e8 0.324155
\(285\) 2.31627e8 0.592697
\(286\) 3.50783e8 0.886660
\(287\) 7.06973e8 1.76529
\(288\) 9.93364e7 0.245039
\(289\) −2.28460e8 −0.556759
\(290\) 1.25436e8 0.302015
\(291\) −9.24477e8 −2.19923
\(292\) 7.78447e7 0.182974
\(293\) 1.98050e8 0.459979 0.229990 0.973193i \(-0.426131\pi\)
0.229990 + 0.973193i \(0.426131\pi\)
\(294\) −7.74973e8 −1.77857
\(295\) 4.46452e8 1.01251
\(296\) 1.72321e7 0.0386204
\(297\) 2.16393e8 0.479287
\(298\) −3.18896e8 −0.698058
\(299\) 1.16306e9 2.51624
\(300\) −6.49137e8 −1.38807
\(301\) 8.25861e8 1.74552
\(302\) −1.71048e7 −0.0357350
\(303\) −2.40452e7 −0.0496569
\(304\) −2.80945e7 −0.0573539
\(305\) −8.02331e8 −1.61921
\(306\) −3.27069e8 −0.652552
\(307\) 3.28496e8 0.647957 0.323979 0.946064i \(-0.394979\pi\)
0.323979 + 0.946064i \(0.394979\pi\)
\(308\) 3.33986e8 0.651329
\(309\) −3.26895e8 −0.630309
\(310\) 7.93191e8 1.51221
\(311\) −5.87246e8 −1.10703 −0.553514 0.832840i \(-0.686713\pi\)
−0.553514 + 0.832840i \(0.686713\pi\)
\(312\) 4.57220e8 0.852283
\(313\) 6.29976e8 1.16123 0.580616 0.814178i \(-0.302812\pi\)
0.580616 + 0.814178i \(0.302812\pi\)
\(314\) −2.43762e8 −0.444337
\(315\) −2.08495e9 −3.75845
\(316\) 2.52508e8 0.450164
\(317\) 8.00885e8 1.41209 0.706046 0.708166i \(-0.250477\pi\)
0.706046 + 0.708166i \(0.250477\pi\)
\(318\) 5.38194e7 0.0938521
\(319\) −1.18971e8 −0.205199
\(320\) 1.22545e8 0.209060
\(321\) −5.30153e8 −0.894609
\(322\) 1.10737e9 1.84840
\(323\) 9.25022e7 0.152737
\(324\) −1.42262e8 −0.232370
\(325\) −1.73566e9 −2.80462
\(326\) −7.59162e8 −1.21359
\(327\) 7.50711e8 1.18729
\(328\) −2.46032e8 −0.384975
\(329\) −7.03905e8 −1.08975
\(330\) 9.58264e8 1.46787
\(331\) −4.54262e8 −0.688507 −0.344253 0.938877i \(-0.611868\pi\)
−0.344253 + 0.938877i \(0.611868\pi\)
\(332\) −2.26722e8 −0.340024
\(333\) 1.02030e8 0.151416
\(334\) −8.82653e8 −1.29622
\(335\) 2.22636e8 0.323548
\(336\) 4.35326e8 0.626076
\(337\) −9.81183e8 −1.39651 −0.698257 0.715847i \(-0.746041\pi\)
−0.698257 + 0.715847i \(0.746041\pi\)
\(338\) 7.20526e8 1.01494
\(339\) −3.85737e8 −0.537764
\(340\) −4.03485e8 −0.556738
\(341\) −7.52313e8 −1.02744
\(342\) −1.66345e8 −0.224863
\(343\) −7.61275e8 −1.01862
\(344\) −2.87406e8 −0.380663
\(345\) 3.17723e9 4.16564
\(346\) 8.48898e8 1.10177
\(347\) 2.69918e8 0.346800 0.173400 0.984852i \(-0.444525\pi\)
0.173400 + 0.984852i \(0.444525\pi\)
\(348\) −1.55071e8 −0.197243
\(349\) 1.09409e9 1.37773 0.688863 0.724892i \(-0.258110\pi\)
0.688863 + 0.724892i \(0.258110\pi\)
\(350\) −1.65255e9 −2.06023
\(351\) 7.54151e8 0.930858
\(352\) −1.16229e8 −0.142042
\(353\) 5.02386e8 0.607891 0.303946 0.952689i \(-0.401696\pi\)
0.303946 + 0.952689i \(0.401696\pi\)
\(354\) −5.51927e8 −0.661257
\(355\) 9.13994e8 1.08429
\(356\) −4.80130e8 −0.564006
\(357\) −1.43333e9 −1.66727
\(358\) 3.90681e8 0.450020
\(359\) −4.89464e8 −0.558329 −0.279165 0.960243i \(-0.590058\pi\)
−0.279165 + 0.960243i \(0.590058\pi\)
\(360\) 7.25578e8 0.819645
\(361\) 4.70459e7 0.0526316
\(362\) 4.41659e7 0.0489336
\(363\) 4.98860e8 0.547401
\(364\) 1.16397e9 1.26499
\(365\) 5.68597e8 0.612040
\(366\) 9.91885e8 1.05749
\(367\) −5.54974e8 −0.586059 −0.293029 0.956103i \(-0.594663\pi\)
−0.293029 + 0.956103i \(0.594663\pi\)
\(368\) −3.85372e8 −0.403100
\(369\) −1.45673e9 −1.50934
\(370\) 1.25868e8 0.129184
\(371\) 1.37012e8 0.139299
\(372\) −9.80585e8 −0.987609
\(373\) −1.29835e9 −1.29542 −0.647708 0.761889i \(-0.724272\pi\)
−0.647708 + 0.761889i \(0.724272\pi\)
\(374\) 3.82690e8 0.378266
\(375\) −2.10319e9 −2.05954
\(376\) 2.44964e8 0.237654
\(377\) −4.14628e8 −0.398532
\(378\) 7.18039e8 0.683796
\(379\) 7.49227e8 0.706929 0.353465 0.935448i \(-0.385003\pi\)
0.353465 + 0.935448i \(0.385003\pi\)
\(380\) −2.05209e8 −0.191847
\(381\) 2.48020e9 2.29746
\(382\) −9.24495e8 −0.848561
\(383\) 2.12814e8 0.193555 0.0967774 0.995306i \(-0.469147\pi\)
0.0967774 + 0.995306i \(0.469147\pi\)
\(384\) −1.51497e8 −0.136535
\(385\) 2.43951e9 2.17867
\(386\) 3.21563e7 0.0284584
\(387\) −1.70171e9 −1.49244
\(388\) 8.19036e8 0.711855
\(389\) 3.13367e8 0.269917 0.134958 0.990851i \(-0.456910\pi\)
0.134958 + 0.990851i \(0.456910\pi\)
\(390\) 3.33965e9 2.85085
\(391\) 1.26885e9 1.07348
\(392\) 6.86583e8 0.575694
\(393\) −1.81266e7 −0.0150641
\(394\) 7.13831e8 0.587975
\(395\) 1.84438e9 1.50578
\(396\) −6.88184e8 −0.556893
\(397\) 1.91013e9 1.53213 0.766066 0.642762i \(-0.222211\pi\)
0.766066 + 0.642762i \(0.222211\pi\)
\(398\) −1.39049e9 −1.10555
\(399\) −7.28980e8 −0.574527
\(400\) 5.75100e8 0.449297
\(401\) −1.27797e9 −0.989726 −0.494863 0.868971i \(-0.664782\pi\)
−0.494863 + 0.868971i \(0.664782\pi\)
\(402\) −2.75234e8 −0.211306
\(403\) −2.62189e9 −1.99547
\(404\) 2.13027e7 0.0160731
\(405\) −1.03912e9 −0.777269
\(406\) −3.94773e8 −0.292757
\(407\) −1.19381e8 −0.0877716
\(408\) 4.98809e8 0.363600
\(409\) 2.92650e8 0.211503 0.105752 0.994393i \(-0.466275\pi\)
0.105752 + 0.994393i \(0.466275\pi\)
\(410\) −1.79708e9 −1.28773
\(411\) 1.83931e9 1.30680
\(412\) 2.89611e8 0.204021
\(413\) −1.40508e9 −0.981465
\(414\) −2.28175e9 −1.58040
\(415\) −1.65603e9 −1.13737
\(416\) −4.05072e8 −0.275870
\(417\) −3.12967e9 −2.11360
\(418\) 1.94633e8 0.130347
\(419\) 3.81573e8 0.253413 0.126706 0.991940i \(-0.459559\pi\)
0.126706 + 0.991940i \(0.459559\pi\)
\(420\) 3.17973e9 2.09420
\(421\) 2.64639e9 1.72849 0.864244 0.503073i \(-0.167797\pi\)
0.864244 + 0.503073i \(0.167797\pi\)
\(422\) −1.18550e9 −0.767906
\(423\) 1.45041e9 0.931751
\(424\) −4.76810e7 −0.0303784
\(425\) −1.89354e9 −1.19650
\(426\) −1.12993e9 −0.708137
\(427\) 2.52511e9 1.56957
\(428\) 4.69686e8 0.289571
\(429\) −3.16753e9 −1.93696
\(430\) −2.09928e9 −1.27330
\(431\) 1.69877e9 1.02203 0.511015 0.859572i \(-0.329270\pi\)
0.511015 + 0.859572i \(0.329270\pi\)
\(432\) −2.49883e8 −0.149123
\(433\) 2.37930e9 1.40845 0.704226 0.709975i \(-0.251294\pi\)
0.704226 + 0.709975i \(0.251294\pi\)
\(434\) −2.49634e9 −1.46585
\(435\) −1.13267e9 −0.659770
\(436\) −6.65089e8 −0.384306
\(437\) 6.45328e8 0.369909
\(438\) −7.02930e8 −0.399717
\(439\) 2.18333e9 1.23167 0.615835 0.787875i \(-0.288819\pi\)
0.615835 + 0.787875i \(0.288819\pi\)
\(440\) −8.48969e8 −0.475125
\(441\) 4.06520e9 2.25708
\(442\) 1.33372e9 0.734658
\(443\) −2.41144e9 −1.31784 −0.658922 0.752212i \(-0.728987\pi\)
−0.658922 + 0.752212i \(0.728987\pi\)
\(444\) −1.55604e8 −0.0843686
\(445\) −3.50699e9 −1.88658
\(446\) 1.10875e7 0.00591783
\(447\) 2.87960e9 1.52495
\(448\) −3.85675e8 −0.202651
\(449\) −6.02088e8 −0.313905 −0.156952 0.987606i \(-0.550167\pi\)
−0.156952 + 0.987606i \(0.550167\pi\)
\(450\) 3.40511e9 1.76152
\(451\) 1.70446e9 0.874923
\(452\) 3.41741e8 0.174066
\(453\) 1.54455e8 0.0780652
\(454\) 1.08117e8 0.0542251
\(455\) 8.50196e9 4.23135
\(456\) 2.53690e8 0.125293
\(457\) −3.28085e9 −1.60798 −0.803989 0.594644i \(-0.797293\pi\)
−0.803989 + 0.594644i \(0.797293\pi\)
\(458\) 2.47906e8 0.120575
\(459\) 8.22750e8 0.397122
\(460\) −2.81485e9 −1.34835
\(461\) −3.50631e9 −1.66685 −0.833426 0.552632i \(-0.813624\pi\)
−0.833426 + 0.552632i \(0.813624\pi\)
\(462\) −3.01586e9 −1.42287
\(463\) −3.85170e9 −1.80351 −0.901757 0.432244i \(-0.857722\pi\)
−0.901757 + 0.432244i \(0.857722\pi\)
\(464\) 1.37384e8 0.0638445
\(465\) −7.16244e9 −3.30351
\(466\) −1.63313e9 −0.747603
\(467\) 4.73318e8 0.215052 0.107526 0.994202i \(-0.465707\pi\)
0.107526 + 0.994202i \(0.465707\pi\)
\(468\) −2.39839e9 −1.08158
\(469\) −7.00682e8 −0.313629
\(470\) 1.78928e9 0.794942
\(471\) 2.20115e9 0.970680
\(472\) 4.88977e8 0.214038
\(473\) 1.99109e9 0.865123
\(474\) −2.28013e9 −0.983410
\(475\) −9.63039e8 −0.412303
\(476\) 1.26985e9 0.539670
\(477\) −2.82315e8 −0.119102
\(478\) −1.80476e9 −0.755826
\(479\) 2.43257e9 1.01132 0.505662 0.862732i \(-0.331248\pi\)
0.505662 + 0.862732i \(0.331248\pi\)
\(480\) −1.10657e9 −0.456704
\(481\) −4.16055e8 −0.170468
\(482\) 3.32925e8 0.135420
\(483\) −9.99941e9 −4.03793
\(484\) −4.41963e8 −0.177185
\(485\) 5.98244e9 2.38113
\(486\) 2.35198e9 0.929409
\(487\) −3.37943e9 −1.32584 −0.662921 0.748689i \(-0.730683\pi\)
−0.662921 + 0.748689i \(0.730683\pi\)
\(488\) −8.78755e8 −0.342294
\(489\) 6.85516e9 2.65116
\(490\) 5.01498e9 1.92567
\(491\) −1.62343e9 −0.618938 −0.309469 0.950910i \(-0.600151\pi\)
−0.309469 + 0.950910i \(0.600151\pi\)
\(492\) 2.22164e9 0.841001
\(493\) −4.52343e8 −0.170021
\(494\) 6.78317e8 0.253156
\(495\) −5.02667e9 −1.86278
\(496\) 8.68744e8 0.319673
\(497\) −2.87653e9 −1.05105
\(498\) 2.04727e9 0.742802
\(499\) 3.88153e9 1.39846 0.699231 0.714896i \(-0.253526\pi\)
0.699231 + 0.714896i \(0.253526\pi\)
\(500\) 1.86331e9 0.666639
\(501\) 7.97027e9 2.83166
\(502\) 3.64463e9 1.28585
\(503\) 2.38583e9 0.835893 0.417947 0.908472i \(-0.362750\pi\)
0.417947 + 0.908472i \(0.362750\pi\)
\(504\) −2.28355e9 −0.794517
\(505\) 1.55600e8 0.0537639
\(506\) 2.66978e9 0.916113
\(507\) −6.50628e9 −2.21720
\(508\) −2.19732e9 −0.743652
\(509\) −3.37295e9 −1.13370 −0.566849 0.823822i \(-0.691838\pi\)
−0.566849 + 0.823822i \(0.691838\pi\)
\(510\) 3.64343e9 1.21623
\(511\) −1.78949e9 −0.593277
\(512\) 1.34218e8 0.0441942
\(513\) 4.18444e8 0.136844
\(514\) −1.97011e9 −0.639912
\(515\) 2.11539e9 0.682441
\(516\) 2.59525e9 0.831582
\(517\) −1.69707e9 −0.540109
\(518\) −3.96132e8 −0.125223
\(519\) −7.66547e9 −2.40687
\(520\) −2.95875e9 −0.922775
\(521\) 5.32384e9 1.64927 0.824637 0.565662i \(-0.191379\pi\)
0.824637 + 0.565662i \(0.191379\pi\)
\(522\) 8.13439e8 0.250310
\(523\) 4.07644e9 1.24602 0.623010 0.782214i \(-0.285910\pi\)
0.623010 + 0.782214i \(0.285910\pi\)
\(524\) 1.60592e7 0.00487601
\(525\) 1.49224e10 4.50070
\(526\) 3.23848e9 0.970268
\(527\) −2.86038e9 −0.851307
\(528\) 1.04954e9 0.310299
\(529\) 5.44713e9 1.59983
\(530\) −3.48274e8 −0.101615
\(531\) 2.89519e9 0.839163
\(532\) 6.45836e8 0.185965
\(533\) 5.94023e9 1.69925
\(534\) 4.33552e9 1.23210
\(535\) 3.43071e9 0.968602
\(536\) 2.43843e8 0.0683964
\(537\) −3.52781e9 −0.983095
\(538\) −1.95668e9 −0.541728
\(539\) −4.75652e9 −1.30837
\(540\) −1.82521e9 −0.498809
\(541\) −3.63310e9 −0.986477 −0.493239 0.869894i \(-0.664187\pi\)
−0.493239 + 0.869894i \(0.664187\pi\)
\(542\) 6.30951e8 0.170215
\(543\) −3.98814e8 −0.106898
\(544\) −4.41918e8 −0.117692
\(545\) −4.85798e9 −1.28549
\(546\) −1.05106e10 −2.76345
\(547\) −5.74113e9 −1.49983 −0.749914 0.661535i \(-0.769905\pi\)
−0.749914 + 0.661535i \(0.769905\pi\)
\(548\) −1.62952e9 −0.422989
\(549\) −5.20303e9 −1.34200
\(550\) −3.98418e9 −1.02110
\(551\) −2.30058e8 −0.0585877
\(552\) 3.47987e9 0.880595
\(553\) −5.80466e9 −1.45962
\(554\) −1.84263e9 −0.460421
\(555\) −1.13657e9 −0.282210
\(556\) 2.77272e9 0.684138
\(557\) 9.95936e8 0.244196 0.122098 0.992518i \(-0.461038\pi\)
0.122098 + 0.992518i \(0.461038\pi\)
\(558\) 5.14376e9 1.25332
\(559\) 6.93917e9 1.68022
\(560\) −2.81707e9 −0.677858
\(561\) −3.45566e9 −0.826344
\(562\) −4.35671e9 −1.03534
\(563\) −4.05283e9 −0.957147 −0.478573 0.878048i \(-0.658846\pi\)
−0.478573 + 0.878048i \(0.658846\pi\)
\(564\) −2.21200e9 −0.519169
\(565\) 2.49617e9 0.582242
\(566\) −2.95910e8 −0.0685965
\(567\) 3.27031e9 0.753440
\(568\) 1.00105e9 0.229212
\(569\) 3.35141e9 0.762666 0.381333 0.924438i \(-0.375465\pi\)
0.381333 + 0.924438i \(0.375465\pi\)
\(570\) 1.85302e9 0.419100
\(571\) 4.22454e9 0.949627 0.474813 0.880087i \(-0.342516\pi\)
0.474813 + 0.880087i \(0.342516\pi\)
\(572\) 2.80626e9 0.626963
\(573\) 8.34810e9 1.85373
\(574\) 5.65579e9 1.24825
\(575\) −1.32100e10 −2.89778
\(576\) 7.94691e8 0.173269
\(577\) −6.43062e9 −1.39360 −0.696799 0.717266i \(-0.745393\pi\)
−0.696799 + 0.717266i \(0.745393\pi\)
\(578\) −1.82768e9 −0.393688
\(579\) −2.90369e8 −0.0621691
\(580\) 1.00349e9 0.213557
\(581\) 5.21188e9 1.10250
\(582\) −7.39581e9 −1.55509
\(583\) 3.30325e8 0.0690402
\(584\) 6.22757e8 0.129382
\(585\) −1.75185e10 −3.61785
\(586\) 1.58440e9 0.325254
\(587\) −2.90754e9 −0.593326 −0.296663 0.954982i \(-0.595874\pi\)
−0.296663 + 0.954982i \(0.595874\pi\)
\(588\) −6.19978e9 −1.25764
\(589\) −1.45476e9 −0.293352
\(590\) 3.57161e9 0.715949
\(591\) −6.44582e9 −1.28447
\(592\) 1.37857e8 0.0273088
\(593\) −5.13346e8 −0.101093 −0.0505463 0.998722i \(-0.516096\pi\)
−0.0505463 + 0.998722i \(0.516096\pi\)
\(594\) 1.73114e9 0.338907
\(595\) 9.27531e9 1.80517
\(596\) −2.55117e9 −0.493602
\(597\) 1.25560e10 2.41513
\(598\) 9.30447e9 1.77925
\(599\) −3.44859e9 −0.655613 −0.327807 0.944745i \(-0.606309\pi\)
−0.327807 + 0.944745i \(0.606309\pi\)
\(600\) −5.19309e9 −0.981515
\(601\) −5.96985e9 −1.12177 −0.560883 0.827895i \(-0.689538\pi\)
−0.560883 + 0.827895i \(0.689538\pi\)
\(602\) 6.60689e9 1.23427
\(603\) 1.44377e9 0.268156
\(604\) −1.36838e8 −0.0252684
\(605\) −3.22821e9 −0.592676
\(606\) −1.92362e8 −0.0351127
\(607\) −1.00863e10 −1.83050 −0.915249 0.402888i \(-0.868007\pi\)
−0.915249 + 0.402888i \(0.868007\pi\)
\(608\) −2.24756e8 −0.0405554
\(609\) 3.56477e9 0.639544
\(610\) −6.41865e9 −1.14496
\(611\) −5.91445e9 −1.04899
\(612\) −2.61656e9 −0.461424
\(613\) 1.24543e9 0.218378 0.109189 0.994021i \(-0.465175\pi\)
0.109189 + 0.994021i \(0.465175\pi\)
\(614\) 2.62797e9 0.458175
\(615\) 1.62275e10 2.81311
\(616\) 2.67188e9 0.460559
\(617\) −5.96038e8 −0.102159 −0.0510794 0.998695i \(-0.516266\pi\)
−0.0510794 + 0.998695i \(0.516266\pi\)
\(618\) −2.61516e9 −0.445696
\(619\) −3.90038e9 −0.660981 −0.330491 0.943809i \(-0.607214\pi\)
−0.330491 + 0.943809i \(0.607214\pi\)
\(620\) 6.34553e9 1.06929
\(621\) 5.73979e9 0.961780
\(622\) −4.69797e9 −0.782787
\(623\) 1.10372e10 1.82874
\(624\) 3.65776e9 0.602655
\(625\) 2.64094e9 0.432692
\(626\) 5.03981e9 0.821115
\(627\) −1.75752e9 −0.284750
\(628\) −1.95010e9 −0.314193
\(629\) −4.53899e8 −0.0727248
\(630\) −1.66796e10 −2.65763
\(631\) 4.64916e9 0.736668 0.368334 0.929694i \(-0.379928\pi\)
0.368334 + 0.929694i \(0.379928\pi\)
\(632\) 2.02007e9 0.318314
\(633\) 1.07050e10 1.67754
\(634\) 6.40708e9 0.998499
\(635\) −1.60498e10 −2.48749
\(636\) 4.30555e8 0.0663635
\(637\) −1.65770e10 −2.54107
\(638\) −9.51772e8 −0.145098
\(639\) 5.92715e9 0.898655
\(640\) 9.80360e8 0.147828
\(641\) 9.04607e9 1.35662 0.678308 0.734778i \(-0.262714\pi\)
0.678308 + 0.734778i \(0.262714\pi\)
\(642\) −4.24122e9 −0.632584
\(643\) 1.11196e10 1.64949 0.824746 0.565503i \(-0.191318\pi\)
0.824746 + 0.565503i \(0.191318\pi\)
\(644\) 8.85893e9 1.30701
\(645\) 1.89563e10 2.78161
\(646\) 7.40018e8 0.108001
\(647\) −3.91564e9 −0.568379 −0.284189 0.958768i \(-0.591724\pi\)
−0.284189 + 0.958768i \(0.591724\pi\)
\(648\) −1.13809e9 −0.164311
\(649\) −3.38754e9 −0.486439
\(650\) −1.38853e10 −1.98316
\(651\) 2.25417e10 3.20223
\(652\) −6.07329e9 −0.858139
\(653\) −1.07233e10 −1.50707 −0.753536 0.657407i \(-0.771653\pi\)
−0.753536 + 0.657407i \(0.771653\pi\)
\(654\) 6.00569e9 0.839539
\(655\) 1.17300e8 0.0163100
\(656\) −1.96826e9 −0.272219
\(657\) 3.68729e9 0.507258
\(658\) −5.63124e9 −0.770572
\(659\) 9.55747e8 0.130090 0.0650450 0.997882i \(-0.479281\pi\)
0.0650450 + 0.997882i \(0.479281\pi\)
\(660\) 7.66611e9 1.03794
\(661\) −1.17758e10 −1.58594 −0.792971 0.609260i \(-0.791467\pi\)
−0.792971 + 0.609260i \(0.791467\pi\)
\(662\) −3.63409e9 −0.486848
\(663\) −1.20433e10 −1.60490
\(664\) −1.81377e9 −0.240433
\(665\) 4.71735e9 0.622046
\(666\) 8.16238e8 0.107067
\(667\) −3.15570e9 −0.411771
\(668\) −7.06122e9 −0.916563
\(669\) −1.00119e8 −0.0129278
\(670\) 1.78109e9 0.228783
\(671\) 6.08786e9 0.777921
\(672\) 3.48261e9 0.442703
\(673\) 5.31058e9 0.671567 0.335783 0.941939i \(-0.390999\pi\)
0.335783 + 0.941939i \(0.390999\pi\)
\(674\) −7.84946e9 −0.987485
\(675\) −8.56564e9 −1.07200
\(676\) 5.76421e9 0.717672
\(677\) 1.06518e10 1.31936 0.659680 0.751547i \(-0.270692\pi\)
0.659680 + 0.751547i \(0.270692\pi\)
\(678\) −3.08589e9 −0.380257
\(679\) −1.88280e10 −2.30813
\(680\) −3.22788e9 −0.393673
\(681\) −9.76290e8 −0.118458
\(682\) −6.01850e9 −0.726512
\(683\) −3.14122e8 −0.0377247 −0.0188624 0.999822i \(-0.506004\pi\)
−0.0188624 + 0.999822i \(0.506004\pi\)
\(684\) −1.33076e9 −0.159002
\(685\) −1.19025e10 −1.41488
\(686\) −6.09020e9 −0.720273
\(687\) −2.23856e9 −0.263403
\(688\) −2.29925e9 −0.269170
\(689\) 1.15122e9 0.134088
\(690\) 2.54178e10 2.94555
\(691\) −1.44774e10 −1.66923 −0.834615 0.550833i \(-0.814310\pi\)
−0.834615 + 0.550833i \(0.814310\pi\)
\(692\) 6.79119e9 0.779066
\(693\) 1.58200e10 1.80568
\(694\) 2.15934e9 0.245224
\(695\) 2.02526e10 2.28841
\(696\) −1.24057e9 −0.139472
\(697\) 6.48056e9 0.724933
\(698\) 8.75269e9 0.974199
\(699\) 1.47470e10 1.63318
\(700\) −1.32204e10 −1.45680
\(701\) 1.22852e10 1.34701 0.673503 0.739185i \(-0.264789\pi\)
0.673503 + 0.739185i \(0.264789\pi\)
\(702\) 6.03321e9 0.658216
\(703\) −2.30850e8 −0.0250603
\(704\) −9.29836e8 −0.100439
\(705\) −1.61570e10 −1.73660
\(706\) 4.01909e9 0.429844
\(707\) −4.89707e8 −0.0521157
\(708\) −4.41542e9 −0.467579
\(709\) −3.48632e9 −0.367371 −0.183686 0.982985i \(-0.558803\pi\)
−0.183686 + 0.982985i \(0.558803\pi\)
\(710\) 7.31195e9 0.766706
\(711\) 1.19606e10 1.24799
\(712\) −3.84104e9 −0.398812
\(713\) −1.99550e10 −2.06176
\(714\) −1.14666e10 −1.17894
\(715\) 2.04976e10 2.09717
\(716\) 3.12545e9 0.318212
\(717\) 1.62968e10 1.65115
\(718\) −3.91571e9 −0.394798
\(719\) 6.30005e9 0.632110 0.316055 0.948741i \(-0.397642\pi\)
0.316055 + 0.948741i \(0.397642\pi\)
\(720\) 5.80463e9 0.579576
\(721\) −6.65758e9 −0.661520
\(722\) 3.76367e8 0.0372161
\(723\) −3.00628e9 −0.295832
\(724\) 3.53328e8 0.0346013
\(725\) 4.70933e9 0.458962
\(726\) 3.99088e9 0.387071
\(727\) −4.41318e9 −0.425973 −0.212986 0.977055i \(-0.568319\pi\)
−0.212986 + 0.977055i \(0.568319\pi\)
\(728\) 9.31179e9 0.894486
\(729\) −1.63768e10 −1.56561
\(730\) 4.54878e9 0.432777
\(731\) 7.57037e9 0.716814
\(732\) 7.93508e9 0.747760
\(733\) 1.15847e10 1.08648 0.543240 0.839577i \(-0.317197\pi\)
0.543240 + 0.839577i \(0.317197\pi\)
\(734\) −4.43979e9 −0.414406
\(735\) −4.52848e10 −4.20675
\(736\) −3.08297e9 −0.285034
\(737\) −1.68930e9 −0.155443
\(738\) −1.16539e10 −1.06727
\(739\) 1.60174e10 1.45995 0.729974 0.683475i \(-0.239532\pi\)
0.729974 + 0.683475i \(0.239532\pi\)
\(740\) 1.00694e9 0.0913467
\(741\) −6.12514e9 −0.553034
\(742\) 1.09609e9 0.0984994
\(743\) 6.54024e9 0.584969 0.292484 0.956270i \(-0.405518\pi\)
0.292484 + 0.956270i \(0.405518\pi\)
\(744\) −7.84468e9 −0.698345
\(745\) −1.86344e10 −1.65108
\(746\) −1.03868e10 −0.915997
\(747\) −1.07392e10 −0.942648
\(748\) 3.06152e9 0.267474
\(749\) −1.07972e10 −0.938908
\(750\) −1.68255e10 −1.45631
\(751\) 1.04818e10 0.903015 0.451508 0.892267i \(-0.350886\pi\)
0.451508 + 0.892267i \(0.350886\pi\)
\(752\) 1.95971e9 0.168047
\(753\) −3.29107e10 −2.80901
\(754\) −3.31702e9 −0.281805
\(755\) −9.99502e8 −0.0845219
\(756\) 5.74431e9 0.483517
\(757\) 4.11773e9 0.345003 0.172501 0.985009i \(-0.444815\pi\)
0.172501 + 0.985009i \(0.444815\pi\)
\(758\) 5.99381e9 0.499874
\(759\) −2.41079e10 −2.00130
\(760\) −1.64167e9 −0.135656
\(761\) −1.90960e10 −1.57071 −0.785357 0.619043i \(-0.787521\pi\)
−0.785357 + 0.619043i \(0.787521\pi\)
\(762\) 1.98416e10 1.62455
\(763\) 1.52891e10 1.24608
\(764\) −7.39596e9 −0.600023
\(765\) −1.91120e10 −1.54344
\(766\) 1.70251e9 0.136864
\(767\) −1.18059e10 −0.944750
\(768\) −1.21197e9 −0.0965448
\(769\) −1.20473e10 −0.955315 −0.477658 0.878546i \(-0.658514\pi\)
−0.477658 + 0.878546i \(0.658514\pi\)
\(770\) 1.95161e10 1.54055
\(771\) 1.77899e10 1.39792
\(772\) 2.57251e8 0.0201232
\(773\) 4.62608e9 0.360234 0.180117 0.983645i \(-0.442352\pi\)
0.180117 + 0.983645i \(0.442352\pi\)
\(774\) −1.36136e10 −1.05531
\(775\) 2.97793e10 2.29805
\(776\) 6.55229e9 0.503358
\(777\) 3.57703e9 0.273558
\(778\) 2.50694e9 0.190860
\(779\) 3.29596e9 0.249805
\(780\) 2.67172e10 2.01586
\(781\) −6.93512e9 −0.520925
\(782\) 1.01508e10 0.759062
\(783\) −2.04622e9 −0.152330
\(784\) 5.49267e9 0.407077
\(785\) −1.42440e10 −1.05096
\(786\) −1.45013e8 −0.0106519
\(787\) 5.71149e9 0.417675 0.208837 0.977950i \(-0.433032\pi\)
0.208837 + 0.977950i \(0.433032\pi\)
\(788\) 5.71065e9 0.415761
\(789\) −2.92432e10 −2.11961
\(790\) 1.47551e10 1.06475
\(791\) −7.85596e9 −0.564393
\(792\) −5.50548e9 −0.393783
\(793\) 2.12168e10 1.51086
\(794\) 1.52810e10 1.08338
\(795\) 3.14488e9 0.221983
\(796\) −1.11239e10 −0.781740
\(797\) −1.15032e10 −0.804848 −0.402424 0.915453i \(-0.631832\pi\)
−0.402424 + 0.915453i \(0.631832\pi\)
\(798\) −5.83184e9 −0.406252
\(799\) −6.45244e9 −0.447517
\(800\) 4.60080e9 0.317701
\(801\) −2.27424e10 −1.56359
\(802\) −1.02237e10 −0.699842
\(803\) −4.31435e9 −0.294043
\(804\) −2.20188e9 −0.149416
\(805\) 6.47078e10 4.37191
\(806\) −2.09751e10 −1.41101
\(807\) 1.76686e10 1.18344
\(808\) 1.70422e8 0.0113654
\(809\) 1.91918e10 1.27437 0.637184 0.770712i \(-0.280099\pi\)
0.637184 + 0.770712i \(0.280099\pi\)
\(810\) −8.31292e9 −0.549612
\(811\) −5.04377e9 −0.332034 −0.166017 0.986123i \(-0.553091\pi\)
−0.166017 + 0.986123i \(0.553091\pi\)
\(812\) −3.15819e9 −0.207010
\(813\) −5.69743e9 −0.371845
\(814\) −9.55047e8 −0.0620639
\(815\) −4.43609e10 −2.87044
\(816\) 3.99047e9 0.257104
\(817\) 3.85023e9 0.247007
\(818\) 2.34120e9 0.149556
\(819\) 5.51343e10 3.50694
\(820\) −1.43766e10 −0.910560
\(821\) 2.85707e10 1.80185 0.900927 0.433972i \(-0.142888\pi\)
0.900927 + 0.433972i \(0.142888\pi\)
\(822\) 1.47144e10 0.924044
\(823\) 1.82564e10 1.14161 0.570803 0.821087i \(-0.306632\pi\)
0.570803 + 0.821087i \(0.306632\pi\)
\(824\) 2.31689e9 0.144265
\(825\) 3.59768e10 2.23066
\(826\) −1.12406e10 −0.694001
\(827\) −1.92073e9 −0.118086 −0.0590428 0.998255i \(-0.518805\pi\)
−0.0590428 + 0.998255i \(0.518805\pi\)
\(828\) −1.82540e10 −1.11751
\(829\) 4.75211e9 0.289698 0.144849 0.989454i \(-0.453730\pi\)
0.144849 + 0.989454i \(0.453730\pi\)
\(830\) −1.32483e10 −0.804239
\(831\) 1.66388e10 1.00582
\(832\) −3.24057e9 −0.195070
\(833\) −1.80848e10 −1.08407
\(834\) −2.50374e10 −1.49454
\(835\) −5.15770e10 −3.06587
\(836\) 1.55707e9 0.0921690
\(837\) −1.29392e10 −0.762728
\(838\) 3.05258e9 0.179190
\(839\) −1.17511e10 −0.686927 −0.343463 0.939166i \(-0.611600\pi\)
−0.343463 + 0.939166i \(0.611600\pi\)
\(840\) 2.54378e10 1.48082
\(841\) −1.61249e10 −0.934782
\(842\) 2.11711e10 1.22223
\(843\) 3.93407e10 2.26176
\(844\) −9.48400e9 −0.542992
\(845\) 4.21032e10 2.40058
\(846\) 1.16033e10 0.658847
\(847\) 1.01598e10 0.574507
\(848\) −3.81448e8 −0.0214808
\(849\) 2.67204e9 0.149853
\(850\) −1.51483e10 −0.846054
\(851\) −3.16656e9 −0.176130
\(852\) −9.03942e9 −0.500728
\(853\) −1.98045e10 −1.09255 −0.546276 0.837605i \(-0.683955\pi\)
−0.546276 + 0.837605i \(0.683955\pi\)
\(854\) 2.02008e10 1.10986
\(855\) −9.72020e9 −0.531856
\(856\) 3.75749e9 0.204757
\(857\) 1.83237e10 0.994445 0.497223 0.867623i \(-0.334353\pi\)
0.497223 + 0.867623i \(0.334353\pi\)
\(858\) −2.53403e10 −1.36964
\(859\) 2.65234e10 1.42775 0.713876 0.700272i \(-0.246938\pi\)
0.713876 + 0.700272i \(0.246938\pi\)
\(860\) −1.67943e10 −0.900361
\(861\) −5.10712e10 −2.72687
\(862\) 1.35901e10 0.722684
\(863\) −1.77522e10 −0.940188 −0.470094 0.882616i \(-0.655780\pi\)
−0.470094 + 0.882616i \(0.655780\pi\)
\(864\) −1.99906e9 −0.105446
\(865\) 4.96045e10 2.60594
\(866\) 1.90344e10 0.995927
\(867\) 1.65038e10 0.860034
\(868\) −1.99707e10 −1.03651
\(869\) −1.39946e10 −0.723423
\(870\) −9.06140e9 −0.466528
\(871\) −5.88737e9 −0.301897
\(872\) −5.32071e9 −0.271745
\(873\) 3.87955e10 1.97348
\(874\) 5.16262e9 0.261566
\(875\) −4.28338e10 −2.16152
\(876\) −5.62344e9 −0.282643
\(877\) 1.22608e9 0.0613792 0.0306896 0.999529i \(-0.490230\pi\)
0.0306896 + 0.999529i \(0.490230\pi\)
\(878\) 1.74667e10 0.870922
\(879\) −1.43070e10 −0.710537
\(880\) −6.79175e9 −0.335964
\(881\) 6.34486e9 0.312612 0.156306 0.987709i \(-0.450041\pi\)
0.156306 + 0.987709i \(0.450041\pi\)
\(882\) 3.25216e10 1.59600
\(883\) −2.24216e10 −1.09598 −0.547992 0.836484i \(-0.684608\pi\)
−0.547992 + 0.836484i \(0.684608\pi\)
\(884\) 1.06697e10 0.519482
\(885\) −3.22513e10 −1.56403
\(886\) −1.92915e10 −0.931856
\(887\) 3.38047e10 1.62647 0.813233 0.581938i \(-0.197705\pi\)
0.813233 + 0.581938i \(0.197705\pi\)
\(888\) −1.24483e9 −0.0596576
\(889\) 5.05120e10 2.41123
\(890\) −2.80559e10 −1.33401
\(891\) 7.88450e9 0.373424
\(892\) 8.87002e7 0.00418454
\(893\) −3.28166e9 −0.154210
\(894\) 2.30368e10 1.07830
\(895\) 2.28291e10 1.06441
\(896\) −3.08540e9 −0.143296
\(897\) −8.40185e10 −3.88688
\(898\) −4.81671e9 −0.221964
\(899\) 7.11391e9 0.326550
\(900\) 2.72409e10 1.24558
\(901\) 1.25593e9 0.0572045
\(902\) 1.36357e10 0.618664
\(903\) −5.96596e10 −2.69633
\(904\) 2.73393e9 0.123083
\(905\) 2.58079e9 0.115740
\(906\) 1.23564e9 0.0552004
\(907\) −1.56762e10 −0.697616 −0.348808 0.937194i \(-0.613413\pi\)
−0.348808 + 0.937194i \(0.613413\pi\)
\(908\) 8.64939e8 0.0383429
\(909\) 1.00905e9 0.0445595
\(910\) 6.80157e10 2.99202
\(911\) 2.35920e10 1.03383 0.516916 0.856036i \(-0.327080\pi\)
0.516916 + 0.856036i \(0.327080\pi\)
\(912\) 2.02952e9 0.0885956
\(913\) 1.25655e10 0.546426
\(914\) −2.62468e10 −1.13701
\(915\) 5.79598e10 2.50123
\(916\) 1.98325e9 0.0852594
\(917\) −3.69169e8 −0.0158100
\(918\) 6.58200e9 0.280807
\(919\) −3.39353e10 −1.44227 −0.721137 0.692793i \(-0.756380\pi\)
−0.721137 + 0.692793i \(0.756380\pi\)
\(920\) −2.25188e10 −0.953428
\(921\) −2.37303e10 −1.00091
\(922\) −2.80505e10 −1.17864
\(923\) −2.41696e10 −1.01173
\(924\) −2.41269e10 −1.00612
\(925\) 4.72554e9 0.196316
\(926\) −3.08136e10 −1.27528
\(927\) 1.37181e10 0.565607
\(928\) 1.09907e9 0.0451449
\(929\) 7.75319e9 0.317267 0.158634 0.987338i \(-0.449291\pi\)
0.158634 + 0.987338i \(0.449291\pi\)
\(930\) −5.72995e10 −2.33593
\(931\) −9.19780e9 −0.373560
\(932\) −1.30651e10 −0.528635
\(933\) 4.24222e10 1.71004
\(934\) 3.78654e9 0.152065
\(935\) 2.23621e10 0.894690
\(936\) −1.91872e10 −0.764795
\(937\) −3.05613e10 −1.21362 −0.606811 0.794846i \(-0.707552\pi\)
−0.606811 + 0.794846i \(0.707552\pi\)
\(938\) −5.60546e9 −0.221769
\(939\) −4.55090e10 −1.79377
\(940\) 1.43142e10 0.562109
\(941\) −4.59802e10 −1.79890 −0.899451 0.437022i \(-0.856033\pi\)
−0.899451 + 0.437022i \(0.856033\pi\)
\(942\) 1.76092e10 0.686374
\(943\) 4.52107e10 1.75570
\(944\) 3.91182e9 0.151348
\(945\) 4.19579e10 1.61734
\(946\) 1.59288e10 0.611735
\(947\) −3.07383e10 −1.17613 −0.588065 0.808813i \(-0.700110\pi\)
−0.588065 + 0.808813i \(0.700110\pi\)
\(948\) −1.82410e10 −0.695376
\(949\) −1.50359e10 −0.571083
\(950\) −7.70431e9 −0.291542
\(951\) −5.78553e10 −2.18128
\(952\) 1.01588e10 0.381604
\(953\) 1.04326e10 0.390451 0.195226 0.980758i \(-0.437456\pi\)
0.195226 + 0.980758i \(0.437456\pi\)
\(954\) −2.25852e9 −0.0842180
\(955\) −5.40220e10 −2.00705
\(956\) −1.44381e10 −0.534449
\(957\) 8.59441e9 0.316974
\(958\) 1.94605e10 0.715114
\(959\) 3.74595e10 1.37150
\(960\) −8.85256e9 −0.322938
\(961\) 1.74720e10 0.635054
\(962\) −3.32844e9 −0.120539
\(963\) 2.22478e10 0.802776
\(964\) 2.66340e9 0.0957562
\(965\) 1.87902e9 0.0673111
\(966\) −7.99953e10 −2.85525
\(967\) 1.29860e10 0.461829 0.230915 0.972974i \(-0.425828\pi\)
0.230915 + 0.972974i \(0.425828\pi\)
\(968\) −3.53570e9 −0.125289
\(969\) −6.68229e9 −0.235935
\(970\) 4.78595e10 1.68371
\(971\) −5.17386e10 −1.81362 −0.906812 0.421536i \(-0.861491\pi\)
−0.906812 + 0.421536i \(0.861491\pi\)
\(972\) 1.88158e10 0.657191
\(973\) −6.37393e10 −2.21826
\(974\) −2.70354e10 −0.937512
\(975\) 1.25383e11 4.33234
\(976\) −7.03004e9 −0.242038
\(977\) 3.29063e10 1.12888 0.564440 0.825474i \(-0.309092\pi\)
0.564440 + 0.825474i \(0.309092\pi\)
\(978\) 5.48413e10 1.87466
\(979\) 2.66100e10 0.906370
\(980\) 4.01198e10 1.36166
\(981\) −3.15035e10 −1.06541
\(982\) −1.29874e10 −0.437655
\(983\) −2.58802e9 −0.0869021 −0.0434510 0.999056i \(-0.513835\pi\)
−0.0434510 + 0.999056i \(0.513835\pi\)
\(984\) 1.77732e10 0.594678
\(985\) 4.17120e10 1.39070
\(986\) −3.61874e9 −0.120223
\(987\) 5.08495e10 1.68336
\(988\) 5.42654e9 0.179008
\(989\) 5.28136e10 1.73603
\(990\) −4.02134e10 −1.31719
\(991\) −4.62321e10 −1.50899 −0.754494 0.656306i \(-0.772118\pi\)
−0.754494 + 0.656306i \(0.772118\pi\)
\(992\) 6.94995e9 0.226043
\(993\) 3.28155e10 1.06355
\(994\) −2.30122e10 −0.743201
\(995\) −8.12520e10 −2.61489
\(996\) 1.63782e10 0.525241
\(997\) 2.54695e10 0.813930 0.406965 0.913444i \(-0.366587\pi\)
0.406965 + 0.913444i \(0.366587\pi\)
\(998\) 3.10522e10 0.988862
\(999\) −2.05326e9 −0.0651577
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 38.8.a.d.1.1 2
3.2 odd 2 342.8.a.g.1.1 2
4.3 odd 2 304.8.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.8.a.d.1.1 2 1.1 even 1 trivial
304.8.a.d.1.2 2 4.3 odd 2
342.8.a.g.1.1 2 3.2 odd 2